162

Ciência eNatura, Santa Maria, v. 37 n. 2 jun. 2015, p. 162−172

ISSN impressa: 0100-8307 ISSN on-line: 2179-460X

Numerical Investigation Of Nanofluids Flow and Heat Transfer in 90

Elbow With square Cross-Section

Ayoob Khosravi Farsani 1, Afshin Ahmadi Nodooshan 2

1. MS student in mechanical engineering, University of Shahrekord, Shahrekord

2. Assistant Professor of Mechanical Engineering, University of Shahrekord, Shahrekord

Abstract

In this article, forced convective heat transfer of nanofluid flow in a horizontal tube with a square cross-

section and 90-degree elbow under heat flux was investigated in a numerical method. The homogeneous

nanofluid of aluminum oxide and water (Al2O3) was used as the working fluid. For the numerical solution of

continuity, momentum and energy equations, the finite volume method was used. In this study, the effect of

Reynolds number and the solid volume fraction and the impact of the elbow on the flow field and the heat

transfer rate and pressure drop was investigated. The results were presented in the form of flow and temperature

contours and the Nusselt diagrams, which have a good relation with the experimental results, and showed that

by increasing the solid volume fraction and Reynolds number, the heat transfer in the elbow increases. Also the

concave surface from inside the tubes had a greater impact on heat transfer than the convex surface.

Keywords: laminar flow, nanofluid, elbow 90 degrees, square cross-section, heat transfer

163

1 INTRODUCTION

Undoubtedly, heat transfer is one of the most

important and most widely used part of

engineering science and its significance can be

multiplied with respect to the energy crisis and

the need for fuel savings. Low thermal

conductivity of conventional heat fluids such as

water, oils, … reduces the efficiency of heating

equipment, so nowadays the new technology of

nanotechnology has been able to help scientists

to introduce the high-capacity thermal

transmission systems called nanofluids.

According to '' the rule of higher thermal

conductivity of fluid compared to metal'', mixed

metal nanoparticles suspended in nanofluid for

increasing thermal conductivity of fluids and

energy saving has taken the focus of

nanomechanical studies.

Here among a lot of researches conducted on

this flows, we refer to several cases. Among the

first theory studies on the curved pipe, we can

point to Mr. Dean research in 1927 [1]. He

indicated that the flow dynamic in bends

depends on Dean dimensionless parameter,

which is in fact the ratio of inertial and

centrifugal forces to viscous force.

Enayet et al [2] achieved speed in a 90-

degrees elbow with square cross-section and

ratio of radius to the channel width of 7 by

Laser-Doppler. They drawn contour speed at

different angles of elbow in the pipe section, and

correspond to Reynolds, they provided average

speed, turbulence intensity and shear stress in

charts to predict the flow, and the results were

used as the basis for turbulence models and

computer software.

Mohammad Boutabaa et al [3] examined

three-dimensional numerical simulation of

secondary flows in developing flow and

Newtonian fluid and viscoelastic in a curved

duct of square cross-section. They conducted

simulation for three different Dean numbers and

using a discrete finite volume and an orthogonal

coordinate system, and showed that in both

Newtonian and non-Newtonian fluids, the size

and numbers of vortices depend on Dean

number, and by increasing the number, Dean

length of development is reduced. The viscosity

also affects on the secondary flow and the

vortices growth rate in viscoelastic fluid is more

than Newtonian fluid.

Taofik et al [4] in an experimental laboratory

studied the characteristics of two nanofluids,

Al2O3/water and CuO/water, in laminar flow in

a square cupric duct and concluded that there is

a significant increase of heat transfer in both

nanofluids compared to base fluid; however

CuO nanofluid showed better heat transfer than

Al2O3 nanofluid.

Sasmito et al [5] in their article examined the

numerical heat transfer by changing flow

geometry and/or by enhancing thermal

conductivity of the fluid. The purpose of this

article was increasing the heat transfer fluid by

simultaneous changing flow geometry and

nanofluid in the quadrilateral cross-section

pipes; and two nanofluids, i.e. water/Al2O3 and

water/CuO have been studied. And also the

amount of thermal conductivity from different

geometries, such as straight pipes, flat-spiral

pipe, cylindrical helix and bevel helical

compared, which in result the heat transfer from

flat-spiral geometry was more than others, as

well as increasing 1 percent of nanofluid

concentration led to a significant increase in heat

transfer, and increasing of the solid particles is

not recommended more than this amount.

Junhong Yang et al [6] experimentally

compared the laminar flow of nanofluid in the

curved and straight pipes and showed that in

the straight pipes, at higher Reynolds numbers

the particles concentration is high, while in the

curved pipes, at higher Reynolds numbers we

have lower concentration of the particles. It

seems that by increasing the Reynolds number,

the effect of bending in stability of suspended

particles is increased; it can be concluded that by

increasing the concentration of particles and

reducing the curvature we can keep fluid in the

higher Reynolds numbers.

Dongsheng et al [7] experimentally

investigated convective heat transfer of Al2 O3

nanofluid flow inside a straight pipe under

laminar flow and concluded that:

164

By using aluminum oxide nanoparticles in

laminar flow, the convective heat transfer is

increased and also the heat transfer is increased

with the increase of the Reynolds number.This

increase in heat transfer is significant especially

in the entrance region and decreases with

distance from the axis; and the length of thermal

developed region in nanofluid is more than the

base fluid, and it increases with the increase of

the particle concentration.Displacement of these

particles reduces the viscosity and reduces the

thickness of the thermal boundary layer which

causes increasing in convective heat transfer.

Alireza Akbari Nia et al [8] numerically

simulated the laminar flow of nanofluid inside

the horizontal pipe in three-dimensional state

with respect to the effects of two centrifugal

and buoyancy forces by using the finite-volume

method. The research results showed that

increasing solid nanoparticles volume has no

effect on the axial velocity and frictional drag

coefficient. Adding solid nanoparticles

increases fluid temperature; it hasn't significant

effects on secondary flow and also increases

heat transfer. Increasing Grashof number

increases secondary flow arised due to

buoyancy force; therefore it leads to disturb the

symmetry in the secondary velocity vectors and

temperature contours and reduces the Nusselt

number.

According to the literature, curved channels

with a quadrilateral cross-section often used in

energy facilities which are widely used in

industrial devices, such as turbomachinery,

cooling systems, air conditioning, heat

exchangers, etc. But little research has been

done in this regard; and since in these sections

the pressure drop and heat transfer are very

important, in this study we decided to examine

laminar flow in a 90-degrees elbow with a

quadrilateral cross- section.

2 Problem Statement

Figure 1 shows the computational region

including curved pipe with axial angle θ

between 0 and 90 degrees in the direction of

constant wall heat flux. The curvature angle of

the pipe is with a radius of 𝑅𝑐 and its

quadrilateral cross-section is to the side of h,

which in the pipe nanofluids of water and

aluminum oxide are flowing slowly and

continuously at the entrance with the speed of

v0 and constant temperature of T0 = 298k. In

this figure L=300 mm ،h=80mm and 𝑅𝑐 =

160 𝑚𝑚. The non-uniform structured grid

shown in Figure 2 is created for discretizing the

computational region. Near the wall due to the

high gradient of speed and temperature the

grid is considered smaller. The sensitivity of the

grid is examined with changing the number of

nodes in different directions.

Figur 1: an overview of the geometry

Figur 2: the computational grid

3 The basic equations governing the nanofluid

flow

165

In this study, it is assumed that the nanofluid

behaves as a homogeneous fluid and the flow of

boundary layer is slow and stable [9] and is

shown that nanofluids behave like the single-

phase fluids [10]. Energy production is

considered zero; and because nanoparticles' size

is extremely small, it seems reasonable that the

mixture can easily flow homogeneously inside

the pipe. So we can ignore sliding motion

between the phases and consider nanofluid as a

continuum environment with thermal

equilibrium between the base fluid and solid

particles. It is assumed that nanofluid is

incompressible with constant physical

properties, which all the properties are

calculated based on inlet temperature fluid as

reference. The work of expansion and viscous

losses in the energy equation are ignored.

The governing equations of laminar flow

inside the channel, assuming incompressible

Newtonian fluid and using the Boussinesq

Approximation are [11]:

(In the above equations subtitles are used as c=

cool surface, f= fluid, h= gram level, m= average,

nf= nanofluids, i =inlet conditions, s= solid, w=

wall).

Continuity equation:

∂u

∂x+

∂v

∂y+

∂w

∂z= 0 (1)

Where

x,y, are the Cartesian coordinates

u,v, and w are velocities in the x and y and (m/s)

z

Momentum equation:

(U∂U

∂X+ V

∂U

∂Y+ W

∂U

∂Z= −

∂P

∂X+

μnf

ρnf

υf

1

Re(

∂2U

∂X2+

∂2U

∂Y2+

∂2U

∂Z2)

(2)

(U∂V

∂X+ V

∂V

∂Y+ W

∂V

∂Z= −

∂P

∂Y+

μnf

ρnf

υf

1

Re(

∂2V

∂X2+

∂2V

∂Y2+

∂2V

∂Z2)

(3)

U∂W

∂X+ V

∂W

∂Y+ W

∂W

∂Z= −

∂P

∂Z+

μnf

ρnf

υf

1

Re(

∂2W

∂X2+

∂2W

∂Y2+

∂2W

∂Z2)

(4)

Where

X,Y,Z are dimensionless coordinates (X = x/L, Y

= y/L, Z = z/L)

U,V,W are dimensionless components of the

velocity (U = u/u_i, V = v/v_i, W = w/ (w_i))

Re is Reynolds number, P = fluid pressure, ρ =

the fluid density (kg m3)⁄ , υ =kinematic viscosity (m2 s⁄ )

Energy equation:

(U∂Θ

∂X+ V

∂Θ

∂Y+ W

∂Θ

∂Z) =

αnf

αf

1

Re.Pr(

∂2Θ

∂X2+

Θ

∂Y2+

∂2Θ

∂Z2) (5)

Where

𝜣 is dimensionless temperature (Θ =T − Tc Th − Tc⁄ )

α is thermal dispersion coefficient of fluid (m2 s⁄ )

Pr is Prandtl number (Pr = υf αf⁄ )

The dimensionless variables used in the

equations are defined as follows:

X =x

L‚ Y =

y

L‚ Z =

z

L‚ U =

u

ui‚ V =

v

vi

W =w

wi‚ Θ =

T − Tc

Th − Tc‚ P =

p̅

ρnf

ui2

Re =ρ

fuiL

μf

‚ Pr =υf

αf

4 Physical and thermal properties of

nanofluids:

The following formulas have been used to study

the physical and thermal properties of

nanofluid:

Nanofluid effective density [12]:

ρnf

= (1 − φ)ρf‚o

+ φρs‚o

(6)

Effective heat capacity of nanofluid:

(Cp)nf = [(1−φ)(ρCp)

f+φ(ρCp)

s

(1−φ)ρf+φρ

s

] (7)

166

The effective thermal conductivity and dynamic

viscosity of nanofluid provided by Ravikanth

and Debendra [13] are as the following

equations:

The effective thermal conductivity of nanofluids:

Knf =ks+2kf−2(kf−ks)

ks+2kf+(kf−ks)φkf +

5 × 104βφρf(Cp)

f√

kT

ρs

dsf(T‚φ) (8)

The effective dynamic viscosity of nanofluids:

𝜇𝑛𝑓 = 5 × 104βφρf√

𝑘𝑏T

ρs

dsf(T‚φ) (9)

f(T‚φ) = (2.8217 × 10−2φ + 3.917 ×

10−3) (T

T0) + (−3.0669 × 10−2φ − 3.91123 ×

10−3) (10)

β = 8.4407(100φ)−1.07304 (11)

1% < 𝜑 < 10%

298k < 𝑇 < 363𝑘

d is diameter of solid particles (20× 10−9) (m),

Cpis the specific heat (j kgk)⁄ , g= acceleration of

gravity (m/s^2), k= thermal conductivity

(w m2k)⁄ , Nu = Nusselt number (Nu =

ql k(Tw − Tc))⁄ , q" = heat flux (w m2)⁄ , Rc =

elbow's radius of curvature(m), L =

dimensionless diameter of pipe, T = temperature

(k), φ = Volume fraction of solid particles (%),

and θ is the angle of curvature (degree)

Table 1: Properties of Water and aluminum oxide particles [14]

Materials 𝛍(𝐩𝐚. 𝐬) 𝐂𝐩(

𝐉

𝐤𝐠𝐤) 𝛒(

𝐤𝐠

𝐦𝟑) 𝐤(

𝐖

𝐦𝐤)

Water

aluminum oxide

0.000993

------

4182

765

998.2

3970

0.613

40

5 Initial conditions and boundary conditions

To get the unique solution of a partial

differential equation, a set of additional

conditions are needed to determine optional

functions resulting from the integration of

partial differential equations. Mentioned

conditions are divided as boundary conditions

and initial conditions. In stability problems

equations merely requires boundary conditions.

In this problem we have used constant wall flux

boundary condition on the pipe walls because it

has many applications in energy facilities; And

also at the entrance we have used Velocity inlet

boundary condition due to incompressible flow

and at outlet we have used Pressure outlet

boundary condition to prevent backward flow.

A list of applied boundary conditions have been

identified in Table 2.

Table 2: boundary conditions

name type Boundary conditions

entrance

outlet

pipe wall

Velocity inlet

Pressure outlet

wall

w=v=0‚ u=1

P = 0

u=v=w=0‚ q = −k(∂T∂r⁄ )

6 Numerical Methods Equations of continuity, momentum and

energy with mentioned boundary conditions

167

were forced according to the finite difference

method based on the finite volume. These

equations were solved by Fluent Software; and

the heat transfer coefficient and the effective

dynamic viscosity were defined by open source

(UDF) written for software. Second order

upwind is used to discrete displacement and

governing equations. The momentum equations

are solved on the displaced grid. In displaced

grid, addition to easily calculate the rate of flow

on volume control, due to given speed on

surface, the amount of pressure is determined

on main point of grid. SIMPLE algorithm has

been used to solve algebraic equations; it is

described in detail in reference [15]. The

convergence criterion is considered as the

residual of velocities which is smaller than 10−6

at each stage. Three-dimensional geometry with

quadrilateral is modeled by Gambit software

and is used for simulation.

7 Verification of written code

To demonstrate the effectiveness of the

method and written program, a comparison has

been carried out between numerical results and

similar works by others. To evaluate computer

program performance developed in problems of

nanofluid convective heat transfer in curved

pipes, a comparison on nanofluid flow in an

elbow was performed according to [16]. In this

investigation, a 180-degree circular elbow is

exposed to the heat flux while a nanofluid is

flowing inside it. In this validation study,

changes of average Nusselt number ratios of the

nanofluid to the base fluid were investigated

with increasing nanoparticles concentration in

different Reynolds numbers. As indicated in Fig.

3a, there is a very small difference between the

results of reference [16] and the results of the

program. To evaluate the performance of the

program in flows inside the quadrilateral cross-

section pipe, its results are compared and

analyzed with done work in reference [17]. In

the above problem the air with temperature of

Tc = 298 enters to a square cross-section 90-

degree elbow and the wall is at the temperature

of Th = 313. In Table 3 the average Nusselt

number rate that obtained from present study is

compared with experimental results of reference

[17]. As you can see, the results are reasonable

and acceptable. (Dean Number (Re(a 2Rc)⁄ 1 2⁄)).

Table 3: Validation

Error

Percent

Average Nusselt

number

Dean

number

Present

work

Refrence

result

6.12

1.94

1.11

2.77

3.62

8.703

11.779

14.13

16.123

17.878

8.17

11.55

14.289

16.584

18.551

200

400

600

800

1000

To select the grid right solution for present

geometry, an investigation was conducted on

the number of grid points. For this purpose, the

effect of grid points' number on average Nusselt

number is determined that indicates the rate of

heat transfer from the hot walls of the channel.

An example of conducted examinations is

shown in Figure 3. In this figure we can see

changes of average Nusselt number of fluid flow

in a square cross-section 90-degrees elbow in

Reynolds 100, based on the number of grid

points. It should be noted that the average

Nusselt number as an affected parameter by the

number of grid points is suitable for this study.

According to Figure 3b, it is clear that almost for

grids which the number of nodes is more than

654360, the answers is remained in the same and

significant changes in values are not seen. Thus,

according to the studies and the implementation

time, a uniform grid with 654000 nodes number

was selected to run the program.

168

Figur 3

8 Results and discussion:

After selecting the appropriate grid and

ensuring the validity of written code, governing

equations were solved with the boundary

conditions. Speed, temperature and pressure are

obtained throughout flow field; and using the

definition of Nusselt number, the heat transfer,

pressure drop and Dimensionless speed are

investigated in different elbow cross-sections at

different Reynolds numbers and different

volume ratio.

Local Nusselt number curves on the inner

wall of elbow for a given amount of heat flux

and constant Reynolds is shown in Fig. 4.

Because of the convex inner wall, the thickness

of the boundary layer begins to increase and the

speed reduces near the wall. For this reason

Nusselt number reduces to the minimum at the

angle of 50 degrees. After the angle of 50-degree

secondary flows are formed and because of this,

after a fluctuation in Nusselt number it

continues its downward trend until the end of

the pipe.

Increasing the volume ratio of solid-liquid at

a constant Reynolds increases the Nusselt

number. Because with the increase of volume

ratio, density increases according to equation

(6), so to keep Reynolds, the speed should be

increased. Thus increasing the solid-liquid

volume ratio with increasing speed increases the

heat transfer and Nusselt number.

Figur 4: the local Nusselt number on the inner

wall of the elbow in Re = 500 and q" = 2500

(w/m2)

In Fig. 5, the curves of local Nusselt number

on the outer wall of elbow for a certain amount

of heat flux and constant Reynolds is shown. It

can be seen that in contrast to the inner wall, the

Nusselt number increases due to a simultaneous

effect of buoyant force and centrifugal force. At

the angle of 50 degrees this upward trend

decreases by forming the secondary flows. But

in 70 degrees gradient gets higher again.

Figur 5: the local Nusselt number on the outer

wall of the elbow in Re = 500 and q" = 2500

(w/m2)

In Fig. 6, the curves of local Nusselt number

on the outer wall of elbow for a given amount of

169

heat flux and constant volume ratio is shown.

As we see, with increasing Reynolds number the

Nusselt number increases because of increasing

speed, but through along the elbow with

reducing the thickness of the boundary layer the

Nusselt number increases due to the centrifugal

force; and in higher Reynolds this upward trend

decrease after angle of 40 degrees because of

forming secondary flows.

Figur 6: the local Nusselt number on the outer

wall of the elbow in

φ = 4% and q" = 1500 (w/m2)

The curves of local Nusselt number on the

inner wall of elbow for a given amount of heat

flux and constant volume ratio is shown. As you

can see in Fig. 7 , by increasing the Reynolds

number like external wall, the Nusselt number

increases; and during the elbow this Nusselt

number at lower Reynolds numbers are

declining due to increasing boundary layer

thickness, and this decline is with a swing at

higher Reynolds because of the formation of

secondary flows.

Figur 7: the local Nusselt number on the inner

wall of the elbow in

φ = 4% and q" = 1500 (w/m2)

The diagram of pressure drop between elbow

input and output at Reynolds numbers and in

different volume ratio is drawn in Figure 8. It is

observed that when the Reynolds number

increases, the pressure drop becomes more

because of increasing secondary flows; and also

by increasing the volume ratio of solid particles

according to equation (6), the density increases

and thus the pressure drop becomes high too.

Figur 8: the elbow pressure drop

Fig. 9, shows the dimensionless axial velocity

profile in cross-section perpendicular to the pipe

at the angle of 90 degrees. As it expected, the

speed on two walls is zero; and because of the

centrifugal force, diagrams have a maximum

value near the outer wall. By increasing the

Reynolds number and the creation of secondary

flow, this maximum value is inclined towards

the center of the pipe.

Figur 9: the dimensionless axial velocity profile

in cross-section perpendicular to the pipe at the

angle of 90 degrees

170

The cantor of flow dimensionless velocity is

drawn in the cross-section of the pipe. Fig. 10

shows how the boundary layer is formed in

input length, secondary flows, return flows in

the elbow and the output length. According to

this figure we can see that the fluid velocity is

growing from input to Output; and at the output

the flow is developed; and speed is higher near

the outer wall.

Figur 10: The cantor of dimensionless velocity in

the cross-section of the pipe in

q" = 2500 (w/m2) and φ = 4% and Re = 500

The contours of fluid dimensionless

temperature at different cross sections of

channel are shown in Fig. 11 (it shows upper

side of the outer wall and lower side of inner

wall). The dimensionless is defined as follows:

𝑇∗ =𝑇−𝑇𝑖

𝑇𝑖 (12)

In the above equation T is the temperature at

any point of fluid and 𝑇𝑖 is input fluid

temperature to the pipe. Referring to the figure

we can see that increasing in temperature occurs

in the thin areanear to outer wall which is as the

result of external heat flux. When the flow

extends from cross- section with the angle θ = 30

° to the cross- section with the angle θ = 90 ° , the

vortex vicinage in the near of inner wall ,so that

the area with high-temperature has transmitted

gradually to the external wall. And also

temperature contours are be changing to the end

of pipe because the constant heat flux is inserted

to the wall to the end of pipe.

𝜃 = 30°

𝜃 = 50°

𝜃 = 70°

𝜃 = 90°

Figur 11: the dimensionless contours at angles of

30, 50, 70 and 90 degrees of elbow q" = 2500

(w/m2) and φ = 4% and Re = 700

Fig. 12, shows the dimensionless velocity

contours (the ratio of speed at each cross-section

to input speed) in sections with different angles

elbow (the upper side is the outer wall and

171

lower side is the inner wall). Due to these

figures, the secondary flow after the angle of 50

degrees begins to form. Because of the low bend,

the flow is not developed in elbow length.

Also high speed area is inclined toward the

outer wall and the low speed area is transferred

to the central part of the cross section. These

changes are due to the centrifugal force.

𝜃 = 30°

𝜃 = 50°

𝜃 = 70°

𝜃 = 90°

Figur 12: the velocity dimensionless contours at

angles of 30, 50, 70 and 90 degrees of elbow

q" = 2500 (w/m2) and φ = 4% and Re = 700

9 Conclusion

laminar and steady flow of nanofluid inside a

pipe with a 90 degree elbow and the square

cross-section were studied with considering the

effects of two buoyancy and centrifugal forces in

the three-dimensional state with numerical

simulation by the Fluent software. It was found

that by increasing the buoyancy force, the

symmetry in the speed and temperature

contours is omitted, while it delays development

of axial velocity. In a given Reynolds number

and constant heat flux, increasing the volume

ratio of solid nanoparticles, increases the Nusselt

number and the fluid temperature but there is

no significant effect on the secondary flows. It

was also observed that the inner wall plays little

role in transferring heat as compared to the

outer wall, that is due to the creation of

secondary flows on the inner wall. It was

observed that increasing two parameters of

Rynoldz number and concentration of

nanoparticles increases the pressure drop due to

increasing secondary flows and density.

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